TOPICS

# Equidistributed Sequence

A sequence of real numbers is equidistributed on an interval if the probability of finding in any subinterval is proportional to the subinterval length. The points of an equidistributed sequence form a dense set on the interval .

However, dense sets need not necessarily be equidistributed. For example, , where is the fractional part, is dense in but not equidistributed, as illustrated above for to 5000 (left) and to (right)

Hardy and Littlewood (1914) proved that the sequence , of power fractional parts is equidistributed for almost all real numbers (i.e., the exceptional set has Lebesgue measure zero). Exceptional numbers include the positive integers, the silver ratio (Finch 2003), and the golden ratio .

The top set of above plots show the values of for equal to e, the Euler-Mascheroni constant , the golden ratio , and pi. Similarly, the bottom set of above plots show a histogram of the distribution of for these constants. Note that while most settle down to a uniform-appearing distribution, curiously appears nonuniform after iterations. Steinhaus (1999) remarks that the highly uniform distribution of has its roots in the form of the continued fraction for .

Now consider the number of empty intervals in the distribution of in the intervals bounded by the intervals determined by 0, , , ..., , 1 for , 2, ..., summarized below for the constants previously considered.

 Sloane # empty intervals for , 2, ... A036412 0, 0, 0, 0, 1, 0, 0, 1, 1, 3, 1, 4, 4, 7, 5, ... A046157 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 3, 0, 3, 5, 3, ... A036414 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, ... A036416 0, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 4, 5, 7, 7, ...

The values of for which no bins are left blank are given in the following table.

 Sloane with no empty intervals A036413 1, 2, 3, 4, 6, 7, 32, 35, 39, 71, 465, 536, 1001, ... A046158 1, 2, 3, 5, 6, 7, 12, 19, 26, 97, 123, 149, 272, 395, ... A036415 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89, 144, ... A036417 1, 6, 7, 106, 112, 113, 33102, 33215, ...

Dense, Kronecker-Weyl Theorem, Normal Number, Power Fractional Parts, Pisot Number, Uniform Distribution, van der Corput's Theorem, Weyl's Criterion

## Explore with Wolfram|Alpha

More things to try:

## References

Hardy, G. H. and Littlewood, J. E. "Some Problems of Diophantine Approximation." Acta Math. 37, 193-239, 1914.Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, 1974.Pólya, G. and Szegö, G. Problems and Theorems in Analysis I. New York: Springer-Verlag, p. 88, 1972.Sloane, N. J. A. Sequences A036412, A036413, A036414, A036415, A036416, A036417, A046157, and A046158 in "The On-Line Encyclopedia of Integer Sequences."Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 155-156, 1991.

## Referenced on Wolfram|Alpha

Equidistributed Sequence

## Cite this as:

Weisstein, Eric W. "Equidistributed Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EquidistributedSequence.html